From the guardian.co.uk article:
Like plastic tips on the end of shoelaces, all eukaryotic species have telomeres on the end of their chromosomes to prevent instability. However, cells seem to struggle to copy telomeres properly when they divide, and very gradually the telomeres become shorter.
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Once a telomere becomes critically short it causes diseases related to chromosomal instability, or limited tissue regeneration, such as cancer and immunodeficiency. “The shortening of telomeres between generations means that eventually the telomeres become critically short for a particular species, causing outbreaks of disease and finally a population crash,” says Stindl. “It could explain the disappearance of a seemingly successful species, like Neanderthal man, with no need for external factors such as climate change.”
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Current estimates suggest telomeres shorten only a tiny amount between each generation, taking thousands of generations to erode to a critical level. Many species can remain stable for tens to hundreds of thousands of years, creating long flat periods in evolution, when nothing much seems to happen.
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Indicators for human extinction

Human telomeres are already relatively short. Are we likely to become extinct soon?

1: Cancer
Cancer incidence does seem to have increased, but it is hard to say whether this is due to longer lifespans, more pollution, or telomere erosion. The shortest telomere in humans occurs on the short arm of chromosome 17; most human cancers are affected by the loss of a tumour suppressor gene on this chromosome.

2: Immunodeficiency
Symptoms of an impaired immune system (like those seen in the Aids patients or the elderly) are related to telomere erosion through immune cells being unable to regenerate. Young people starting to suffer more from diseases caused by an impaired immune system might be a result of telomere shortening between generations.